WASHINGTON—The Supreme Court on Friday freed a group of Roman Catholic nonprofit organizations from complying with the Affordable Care Act's contraceptive provisions while they pursue a challenge to the law before a federal appeals court.

The 2010 federal health-care law requires that plans include contraceptive coverage. Nonprofit employers that claim religious objections to contraception can opt out of paying for their employees to receive such benefits by filing a form with the Labor Department. Insurers generally must...